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  "body": "\n## Definition\n\nLet $M = (E, \\mathcal{I})$ be a [[matroid|PX31ZNYS3Z]] with [[rank function|SYRCTH853H]] $r$, and let $X \\subseteq E$.\n\n1. The *deletion* $M \\setminus X$ is the matroid on $E \\setminus X$ whose independent sets are the sets $I \\in \\mathcal{I}$ with $I \\subseteq E \\setminus X$.\n2. The *contraction* $M / X$ is the matroid on $E \\setminus X$ with rank function $r_{M/X}(A) = r(A \\cup X) - r(X)$ for $A \\subseteq E \\setminus X$.\n3. A matroid $N$ is a *minor* of $M$ if $N = (M \\setminus X) / Y$ for some disjoint sets $X, Y \\subseteq E$.\n\n## Remarks\n\nIf $J$ is a maximal independent subset of $X$, then a set $I \\subseteq E \\setminus X$ is independent in $M / X$ if and only if $I \\cup J \\in \\mathcal{I}$.\nDeletion and contraction commute, and $M / X = (M^* \\setminus X)^*$, where $M^*$ is the [[dual matroid|6QF83CFK2G]].\nMany authors call $N$ a minor when it is isomorphic to a matroid of the form in clause 3.\nFor the graphic matroid of a graph, deleting or contracting an edge gives the graphic matroid of the graph with that edge deleted or contracted.\n\nExample: let $M = U_{2,4}$, the matroid on $\\set{1, 2, 3, 4}$ whose independent sets are the subsets with at most $2$ elements.\nThen $M \\setminus \\set{4} = U_{2,3}$, and $M / \\set{4} = U_{1,3}$, since $r(A \\cup \\set{4}) - 1 = \\min(\\abs{A} + 1, 2) - 1 = \\min(\\abs{A}, 1)$.\n\nNon-example: $U_{2,4}$ is not a minor of $U_{1,5}$, because deletion and contraction never raise the rank, and $U_{2,4}$ has rank $2$ while $U_{1,5}$ has rank $1$.\n\nA theorem of Tutte states that a matroid is representable over the field with two elements if and only if it has no minor isomorphic to $U_{2,4}$.\n",
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